To solve (REF ), we substitute the quadratic upper-bound [1]} for the expected excessive demand from slice \(s\)  given in (REF ), which is convex with a Lipschitz continuous gradient. The Lipschitz constant for the gradient of (REF ) is \(\sum _{k \in \mathcal {K}_s}|\mathcal {P}_k|\) . Similarly, we substitute a quadratic upper-bound for (REF ). If \(Q\)  is the Lipschitz constant for the gradient of (REF ) (\(Q\)  can be calculated through differentiation of (REF ) and using extremum values of the first and second derivatives of \(\phi _{k,S_n}(\cdot )\) ), then (REF ) is an upper-bound for \(\Gamma (\mathbf {r})\)  around the current iterate \(\hat{\mathbf {r}}\) . In (REF ), \(\Upsilon _1\)  and \(\Upsilon _2\)  are constants and we consider \(\Upsilon =\Upsilon _1+\Upsilon _2\) . We notice that (REF ) can be written in the following shorter form:
\(&\sum _{j=1}^{J}\sum _{s \in \mathcal {S}_j}(\frac{\psi _jQ}{2}+\frac{\sum _{k \in \mathcal {K}_s}|\mathcal {P}_k|}{2})\sum _{k \in \mathcal {K}_s}\sum _{p\in \mathcal {P}_{k}}(r_k^{p}-\hat{r}_k^{p})^2\nonumber \\&+\sum _{j=1}^{J}\sum _{s \in \mathcal {S}_j}\sum _{k \in \mathcal {K}_s}\sum _{p\in \mathcal {P}_{k}}h_{k}^p(r_k^{p}-\hat{r}_k^{p})+\Upsilon ,\) 
